Assuming standard regularity conditions [e.g. [1]}], Fisher information in \(\mathbf {X}_1\)  conditional on the interim decision variable \(D\)  is the variance of the score function for
\(f_{\mathbf {X}_1|D}(y_1|D)\) :
\({\cal {I}}_{\mathbf {X}_1|D}(\theta ) &=\textrm {Var} \left[\frac{\partial }{\partial \theta }\log f_{\mathbf {X}_1|D}(y_1|D)\right]\\&= \int _{-\infty }^{\infty } -\frac{\partial ^2}{\partial \theta ^2} [\log f_{\mathbf {X}_1|D}(y_1|D)] f_{\mathbf {X}_1}(\mathbf {x}_1) d\mathbf {x}_1. \) 
